Let us set some global options for all code chunks in this document.

# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = TRUE,    
  # Disable warnings printed by R code chunks
  warning = TRUE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
# Start the figure counter
fig_count <- 0
# Define the captioner function
captioner <- function(caption) {
  fig_count <<- fig_count + 1
  paste0("Figure ", fig_count, ": ", caption)
}
# Define the function to truncate a number to two decimal places
truncate_to_two <- function(x) {
  floor(x * 100) / 100
}

m1table <- rSPDE:::m1table
m2table <- rSPDE:::m2table
m3table <- rSPDE:::m3table
m4table <- rSPDE:::m4table
# install.packages("INLA",repos=c(getOption("repos"),INLA="https://inla.r-inla-download.org/R/testing"), dep=TRUE)
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)
## Loading required package: Matrix
## This is INLA_25.05.01 built 2025-05-01 18:43:33 UTC.
##  - See www.r-inla.org/contact-us for how to get help.
##  - List available models/likelihoods/etc with inla.list.models()
##  - Use inla.doc(<NAME>) to access documentation
##  - Consider upgrading R-INLA to testing[25.05.07] or stable[24.12.11].
library(inlabru)
## Loading required package: fmesher
library(rSPDE)
## This is rSPDE 2.5.1
## - See https://davidbolin.github.io/rSPDE for vignettes and manuals.
library(MetricGraph)
## This is MetricGraph 1.4.1
## - See https://davidbolin.github.io/MetricGraph for vignettes and manuals.
## 
## Attaching package: 'MetricGraph'
## The following object is masked from 'package:stats':
## 
##     filter
library(grateful)

library(ggplot2)
library(reshape2)
library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout

Utilitary functions

# For each m and beta, this function returns c_m/b_{m+1} and the roots of rb and rc
my.get.roots <- function(order, beta) {
  mt <- get(paste0("m", order, "table"))
  rb <- rep(0, order + 1)
  rc <- rep(0, order)
  if(order == 1) {
      rc = approx(mt$beta, mt[[paste0("rc")]], beta)$y
    } else {
      rc = sapply(1:order, function(i) {
        approx(mt$beta, mt[[paste0("rc.", i)]], beta)$y
      })
    }
    rb = sapply(1:(order+1), function(i) {
      approx(mt$beta, mt[[paste0("rb.", i)]], xout = beta)$y
    })
    factor = approx(mt$beta, mt$factor, xout = beta)$y
  return(list(rb = rb, rc = rc, factor = factor))
}

# Function the polynomial coefficients in increasing order like a+bx+cx^2+...
poly.from.roots <- function(roots) {
  
  coef <- 1
  for (r in roots) {coef <- convolve(coef, c(1, -r), type = "open")}
  return(coef)
}
# Function to compute the partial fraction parameters
compute.partial.fraction.param <- function(factor, pr_roots, pl_roots, cte) {
  
  pr_coef <- c(0, poly.from.roots(pr_roots)) 
  pl_coef <- poly.from.roots(pl_roots) 
  factor_pr_coef <- pr_coef
  pr_plus_pl_coef <- factor_pr_coef + cte/factor * pl_coef
  res <- gsignal::residue(factor_pr_coef, pr_plus_pl_coef)
  return(list(r = res$r, p = res$p, k = res$k)) 
}
# Function to compute the fractional operator
my.fractional.operators.frac <- function(L, beta, C, scale.factor, m = 1, time_step) {
  
  C <- Matrix::Diagonal(dim(C)[1], rowSums(C)) 
  Ci <- Matrix::Diagonal(dim(C)[1], 1 / rowSums(C)) 
  I <- Matrix::Diagonal(dim(C)[1])
  L <- L / scale.factor 
  LCi <- L %*% Ci
  if(beta == 1){
    L <- L * scale.factor^beta
    return(list(Ci = Ci, C = C, LCi = LCi, L = L, m = m, beta = beta, LHS = C + time_step * L))
  } else {
    roots <- my.get.roots(m, beta)
    scaling <- scale.factor^beta
    poles_rs_k <- compute.partial.fraction.param(roots$factor, roots$rc, roots$rb, time_step)

    partial_fraction_terms <- list()
    for (i in 1:(m+1)) {partial_fraction_terms[[i]] <- (LCi - poles_rs_k$p[i] * I)/poles_rs_k$r[i]}
    partial_fraction_terms[[m+2]] <- ifelse(is.null(poles_rs_k$k), 0, poles_rs_k$k) * I
    return(list(Ci = Ci, C = C, LCi = LCi, L = L, m = m, beta = beta, partial_fraction_terms = partial_fraction_terms))
  }
}
# Function to solve the iteration
my.solver.frac <- function(obj, v){
  beta <- obj$beta
  m <- obj$m
  C <- obj$C
  Ci <- obj$Ci
  if (beta == 1){
    return(solve(obj$LHS, v))
  } else {
    partial_fraction_terms <- obj$partial_fraction_terms
    output <- partial_fraction_terms[[m+2]] %*% v
    for (i in 1:(m+1)) {output <- output + solve(partial_fraction_terms[[i]], v)}
    return(Ci %*% output)
  }
}
# Function to build a tadpole graph and create a mesh
gets.graph.tadpole <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 10000)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges <- list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$set_manual_edge_lengths(edge_lengths = c(1,2))
  graph$build_mesh(h = h)
  return(graph)
}
# Function to compute the eigenfunctions 
tadpole.eig <- function(k,graph){
x1 <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[graph$mesh$PtE[,1]==1,2]) 
x2 <- c(0,graph$get_edge_lengths()[2]*graph$mesh$PtE[graph$mesh$PtE[,1]==2,2]) 

if(k==0){ 
  f.e1 <- rep(1,length(x1)) 
  f.e2 <- rep(1,length(x2)) 
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  f = list(phi=f1/sqrt(3)) 
  
} else {
  f.e1 <- -2*sin(pi*k*1/2)*cos(pi*k*x1/2) 
  f.e2 <- sin(pi*k*x2/2)                  
  
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  
  if((k %% 2)==1){ 
    f = list(phi=f1/sqrt(3)) 
  } else { 
    f.e1 <- (-1)^{k/2}*cos(pi*k*x1/2)
    f.e2 <- cos(pi*k*x2/2)
    f2 = c(f.e1[1],f.e2[1],f.e1[-1],f.e2[-1]) 
    f <- list(phi=f1,psi=f2/sqrt(3/2))
  }
}
return(f)
}
# Function to order the vertices for plotting
plotting.order <- function(v, graph){
  edge_number <- graph$mesh$VtE[, 1]
  pos <- sum(edge_number == 1)+1
  return(c(v[1], v[3:pos], v[2], v[(pos+1):length(v)], v[2]))
}
# Function to plot in 3D
graph.plotter.3d <- function(graph, 
                             U_true, 
                             U_approx1, 
                             #U_approx2, 
                             time_seq, 
                             time_step){
  x <- graph$mesh$V[, 1]
  y <- graph$mesh$V[, 2]
  x <- plotting.order(x, graph)
  y <- plotting.order(y, graph)
  weights <- graph$mesh$weights
  
  cumsum1 <- sqrt(time_step * cumsum(t(weights) %*% (U_true - U_approx1)^2))
  #cumsum2 <- sqrt(time_step * cumsum(t(weights) %*%(U_true - U_approx2)^2))
  #cumsum3 <- sqrt(time_step * cumsum(t(weights) %*% (U_approx1 - U_approx2)^2))
  
  U_true <- apply(U_true, 2, plotting.order, graph = graph)
  U_approx1 <- apply(U_approx1, 2, plotting.order, graph = graph)
  #U_approx2 <- apply(U_approx2, 2, plotting.order, graph = graph)
  
  plot_data <- data.frame(
  x = rep(x, times = ncol(U_true)),
  y = rep(y, times = ncol(U_true)),
  z_true = as.vector(U_true),
  z_approx1 = as.vector(U_approx1),
  #z_approx2 = as.vector(U_approx2),
  frame = rep(time_seq, each = length(x)))
  
  x_range <- range(x)
  y_range <- range(y)
  z_range <- range(c(U_true, 
                     #U_approx2, 
                     U_approx1))
  
  p <- plot_ly(plot_data, frame = ~frame) %>%
  add_trace(x = ~x, y = ~y, z = ~z_true, type = "scatter3d", mode = "lines", name = "T",
            line = list(color = "green", width = 2)) %>%
  add_trace(x = ~x, y = ~y, z = ~z_approx1, type = "scatter3d", mode = "lines", name = "A1",
            line = list(color = "red", width = 2)) %>%
  # add_trace(x = ~x, y = ~y, z = ~z_approx2, type = "scatter3d", mode = "lines", name = "A2", 
  #           line = list(color = "blue", width = 2)) %>%
  layout(
    scene = list(
      xaxis = list(title = "x", range = x_range),
      yaxis = list(title = "y", range = y_range),
      zaxis = list(title = "z", range = z_range),
      aspectratio = list(x = 2.4, y = 1.2, z = 1.2),
      camera = list(eye = list(x = 1.5, y = 1.5, z = 1), center = list(x = 0, y = 0, z = 0))),
    updatemenus = list(list(type = "buttons", showactive = FALSE,
                            buttons = list(
          list(label = "Play", method = "animate",
               args = list(NULL, list(frame = list(duration = 100, redraw = TRUE), fromcurrent = TRUE))),
          list(label = "Pause", method = "animate",
               args = list(NULL, list(mode = "immediate", frame = list(duration = 0), redraw = FALSE)))
        )
      )
    ),
    title = "Time: 0"
  ) %>% plotly_build()
  for (i in seq_along(p$x$frames)) {
    t <- time_seq[i]
    err1 <- signif(cumsum1[i], 6)
    #err2 <- signif(cumsum2[i], 4)
    #err3 <- signif(cumsum3[i], 4)
    p$x$frames[[i]]$layout <- list(title = paste0("Time: ", t, 
                                                  # " | cum E=T-A2: ", err2, 
                                                  # " | cum E=A1-A2: ", err3,
                                                  " | cum E=T-A1: ", err1))
  }
  return(p)
}
# Function to plot the error at each time step
error.at.each.time.plotter <- function(graph, U_true, U_approx1, U_approx2, time_seq, time_step) {
  weights <- graph$mesh$weights
  error_at_each_time1 <- t(weights) %*% (U_true - U_approx1)^2
  error_at_each_time2 <- t(weights) %*% (U_true - U_approx2)^2
  error_between_both_approx <- t(weights) %*% (U_approx1 - U_approx2)^2
  
  error1 <- sqrt(as.double(t(weights) %*% (U_true - U_approx1)^2 %*% rep(time_step, ncol(U_true))))
  error2 <- sqrt(as.double(t(weights) %*% (U_true - U_approx2)^2 %*% rep(time_step, ncol(U_true))))
  errorb <- sqrt(as.double(t(weights) %*% (U_approx1 - U_approx2)^2 %*% rep(time_step, ncol(U_true))))
  
  fig <- plot_ly() %>% 
  add_trace(
  x = ~time_seq, y = ~error_at_each_time1, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'red', width = 2),
  marker = list(color = 'red', size = 4),
  name = paste0("E=T-A1: ", sprintf("%.3e", error1))
) %>% 
  add_trace(
  x = ~time_seq, y = ~error_at_each_time2, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'blue', width = 2, dash = "dot"),
  marker = list(color = 'blue', size = 4),
  name = paste0("E=T-A2: ", sprintf("%.3e", error2))
) %>% 
  add_trace(
  x = ~time_seq, y = ~error_between_both_approx, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'orange', width = 2),
  marker = list(color = 'orange', size = 4),
  name = paste0("E=A1-A2: ", sprintf("%.3e", errorb))
) %>% 
  layout(
  title = "Error at Each Time Step",
  xaxis = list(title = "Error at Each Time Step"),
  yaxis = list(title = "Error"),
  legend = list(x = 0.1, y = 0.9)
)
  return(fig)
}
# Function to compute the eigenvalues and eigenfunctions
gets.eigen.params <- function(N_finite = 4, kappa = 1, alpha = 0.5, graph){
  EIGENVAL_ALPHA <- NULL
  EIGENFUN <- NULL
  for (j in 0:N_finite) {
      lambda_j_alpha <- (kappa^2 + (j*pi/2)^2)^(alpha/2)
      e_j <- tadpole.eig(j,graph)$phi
      EIGENVAL_ALPHA <- c(EIGENVAL_ALPHA, lambda_j_alpha)         
      EIGENFUN <- cbind(EIGENFUN, e_j)
      if (j>0 && (j %% 2 == 0)) {
        lambda_j_alpha <- (kappa^2 + (j*pi/2)^2)^(alpha/2)
        e_j <- tadpole.eig(j,graph)$psi
        EIGENVAL_ALPHA <- c(EIGENVAL_ALPHA, lambda_j_alpha)         
        EIGENFUN <- cbind(EIGENFUN, e_j)
      }
  }
  return(list(EIGENVAL_ALPHA = EIGENVAL_ALPHA,
              EIGENFUN = EIGENFUN))
}

We want to solve the fractional diffusion equation \[\begin{equation} \label{eq:maineq} \partial_t u+(\kappa^2-\Delta_\Gamma)^{\alpha/2} u=f \text { on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma, \end{equation}\] where \(u\) satisfies the Kirchhoff vertex conditions \[\begin{equation} \label{eq:Kcond} \left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V: \sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\} \end{equation}\] The solution is given by \[\begin{equation} \label{eq:sol_reprentation} u(s,t) = \displaystyle\sum_{j\in\mathbb{N}}e^{-\lambda^{\alpha/2}_jt}\left(u_0, e_j\right)_{L_2(\Gamma)}e_j(s) + \int_0^t \displaystyle\sum_{j\in\mathbb{N}}e^{-\lambda^{\alpha/2}_j(t-r)}\left(f(\cdot, r), e_j\right)_{L_2(\Gamma)}e_j(s)dr. \end{equation}\]

If we choose \(w_j\) and \(v_j\) and take the initial condition and the right hand side funciton as

\[\begin{equation} u_0(s) = \sum_{j=0}^{N} w_j e_j(s) \text{ and so } \left(u_0, e_j\right)_{L_2(\Gamma)} = w_j, \end{equation}\] \[\begin{equation} \text{In matrix notation: } \quad\boldsymbol{U}_0 = \boldsymbol{E}_h\boldsymbol{c}, \quad \boldsymbol{E}_h = \left[e_0, e_1, \ldots, e_{N}\right], \quad \boldsymbol{w} = \left[w_0, w_1, \ldots, w_{N}\right]^\top, \end{equation}\] \[\begin{equation} \text{In } \texttt{R}: \texttt{U_0 <- EIGENFUN %*% coeff_U_0} \end{equation}\] and \[\begin{equation} f(s,t) = \sum_{j=0}^{N} v_j e^{-\lambda^{\alpha/2}_jt} e_j(s) \text{ and so } \left(f(\cdot,r), e_j\right)_{L_2(\Gamma)} = v_j e^{-\lambda^{\alpha/2}_jr}, \end{equation}\] \[\begin{equation} \text{In matrix notation: } \quad\boldsymbol{f} = \boldsymbol{E}_h \boldsymbol{V}, \quad \boldsymbol{V}_{ji} = v_je^{-\lambda^{\alpha/2}_jt_i} \end{equation}\] \[\begin{equation} \text{In } \texttt{R}: \texttt{FF <- EIGENFUN %*% (coeff_FF * exp(-outer(EIGENVAL_ALPHA, time_seq)))} \end{equation}\] then the solution is given by \[\begin{align} u(s,t) &= \displaystyle\sum_{j=0}^{N}w_je^{-\lambda^{\alpha/2}_jt}e_j(s) + \int_0^t \displaystyle\sum_{j=0}^Ne^{-\lambda^{\alpha/2}_j(t-r)}v_j e^{-\lambda^{\alpha/2}_jr}e_j(s)dr\\ &= \displaystyle\sum_{j=0}^{N}w_je^{-\lambda^{\alpha/2}_jt}e_j(s) + t \displaystyle\sum_{j=0}^Nv_j e^{-\lambda^{\alpha/2}_jt}e_j(s)\\ &= \displaystyle\sum_{j=0}^{N}w_je^{-\lambda^{\alpha/2}_jt}e_j(s) + tf(s,t)\\ &= \displaystyle\sum_{j=0}^{N}(w_j+tv_j)e^{-\lambda^{\alpha/2}_jt}e_j(s) \end{align}\]

\[\begin{equation} \text{In matrix notation: } \quad\boldsymbol{U} =\boldsymbol{E}_h \boldsymbol{W} + (\boldsymbol{t}\boldsymbol{f}^\top)^\top, \quad \boldsymbol{W}_{ji} = w_je^{-\lambda^{\alpha/2}_jt_i},\quad \boldsymbol{t} = \left[t_0, t_1, \ldots, t_{K}\right]^\top \end{equation}\] \[\begin{equation} \text{In } \texttt{R}: \texttt{U_true <- EIGENFUN %*% (coeff_U_0 * exp(-outer(EIGENVAL_ALPHA, time_seq)))+ t(time_seq * t(FF))} \end{equation}\]

kappa <- 1
alpha <- 0.5 # from 0.5 to 2
m = 4
beta <- alpha/2
N_finite = 4 # choose even
adjusted_N_finite <- N_finite + N_finite/2 + 1
# Coefficients for u_0 and f
coeff_U_0 <- 50*(1:adjusted_N_finite)^-1
coeff_U_0[-5] <- 0
coeff_FF <- rep(0, adjusted_N_finite)
coeff_FF[7] <- 10
T_final <- 2

overkill_time_power <- 9
overkill_h_power <- 9

time_steps <- 0.1 * 2^-c(overkill_time_power:0)
hs <- 0.1 * 2^-c(overkill_h_power:0)

overkill_time_step <- time_steps[1]
overkill_h <- hs[1]

overkill_time_seq <- seq(0, T_final, by = overkill_time_step)

overkill_graph <- gets.graph.tadpole(h = overkill_h)
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.17 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
overkill_graph$compute_fem()
overkill_weights <- overkill_graph$mesh$weights
overkill_eigen_params <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = overkill_graph)
EIGENVAL_ALPHA <- overkill_eigen_params$EIGENVAL_ALPHA
overkill_EIGENFUN <- overkill_eigen_params$EIGENFUN


overkill_U_true <- overkill_EIGENFUN %*% 
  outer(1:length(coeff_U_0), 1:length(overkill_time_seq), 
        function(i, j) (coeff_U_0[i] + coeff_FF[i] * overkill_time_seq[j]) * exp(-EIGENVAL_ALPHA[i] * overkill_time_seq[j]))

by_vector <- 2^c(0:overkill_time_power)

errors <- matrix(NA, nrow = length(hs), ncol = length(time_steps))
for (i in 1:length(hs)) {
  h <- hs[i]
  graph <- gets.graph.tadpole(h = h)
  graph$compute_fem()
  G <- graph$mesh$G
  C <- graph$mesh$C
  L <- kappa^2*C + G
  eigen_params <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = graph)
  EIGENFUN <- eigen_params$EIGENFUN
  U_0 <- EIGENFUN %*% coeff_U_0
  A <- graph$fem_basis(overkill_graph$get_mesh_locations())
  for (j in 1:length(time_steps)) {
    time_step <- time_steps[j]
    coarse_indices <- seq(1, length(overkill_time_seq), by = by_vector[j])
    time_seq <- overkill_time_seq[coarse_indices]
    my_op_frac <- my.fractional.operators.frac(L, beta, C, scale.factor = kappa^2, m = m, time_step)
    INT_BASIS_EIGEN <- t(overkill_EIGENFUN) %*% overkill_graph$mesh$C %*% A
    FF_approx <- t(INT_BASIS_EIGEN) %*% 
      outer(1:length(coeff_FF), 1:length(time_seq), 
        function(i, j) coeff_FF[i] * exp(-EIGENVAL_ALPHA[i] * time_seq[j]))
    U_approx <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
    U_approx[, 1] <- U_0
    for (k in 1:(length(time_seq) - 1)) {
      U_approx[, k + 1] <- as.matrix(my.solver.frac(my_op_frac, my_op_frac$C %*% U_approx[, k] + time_step * FF_approx[, k + 1]))
      }
    sliced_overkill_U_true <- overkill_U_true[, coarse_indices]
    projected_U_approx <- A %*% U_approx
    errors[i,j] <- sqrt(as.double(t(overkill_weights) %*% (sliced_overkill_U_true - projected_U_approx)^2 %*% rep(time_step, length(time_seq))))
  }
}
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.16 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.28 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.35 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.35 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.15 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.47 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.18 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.22 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.21 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.28 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
graph.plotter.3d(overkill_graph, sliced_overkill_U_true, projected_U_approx, time_seq, time_step)
df <- melt(errors)
colnames(df) <- c("h", "time_step", "Value")

df$h <- factor(df$h, levels = unique(df$h), labels = hs)
df$time_step <- factor(df$time_step, levels = unique(df$time_step), labels = time_steps)

p <- ggplot(df, aes(x = time_step, y = h, fill = Value, text = paste("Value:", Value))) +
  geom_tile(color = "black") +  # Add borders
  geom_text(aes(label = round(Value, 8)), size = 3) +  # Add value labels
  scale_fill_gradient(low = "white", high = "red") +
  coord_fixed() +
  theme_minimal()

ggplotly(p, tooltip = "text")

Figure 1: Caption

print(errors)
##              [,1]        [,2]        [,3]        [,4]       [,5]       [,6]
##  [1,] 0.001671123 0.002121036 0.003465794 0.006588292 0.01309159 0.02619082
##  [2,] 0.001670331 0.002120388 0.003465369 0.006588037 0.01309143 0.02619071
##  [3,] 0.001667170 0.002117807 0.003463675 0.006587023 0.01309079 0.02619024
##  [4,] 0.001654636 0.002107583 0.003456974 0.006583005 0.01308824 0.02618839
##  [5,] 0.001606342 0.002068369 0.003431356 0.006567599 0.01307839 0.02618115
##  [6,] 0.001447460 0.001941109 0.003348601 0.006516732 0.01304448 0.02615496
##  [7,] 0.001571469 0.002010531 0.003359244 0.006490142 0.01299742 0.02609443
##  [8,] 0.006399347 0.006490531 0.006968731 0.008817213 0.01417955 0.02655839
##  [9,] 0.027981299 0.027974735 0.028033767 0.028436497 0.03030460 0.03731898
## [10,] 0.113710063 0.113685061 0.113652086 0.113654166 0.11392904 0.11553968
##             [,7]      [,8]      [,9]     [,10]
##  [1,] 0.05227199 0.1037405 0.2039268 0.3942324
##  [2,] 0.05227188 0.1037404 0.2039267 0.3942322
##  [3,] 0.05227148 0.1037400 0.2039262 0.3942315
##  [4,] 0.05226987 0.1037383 0.2039241 0.3942285
##  [5,] 0.05226351 0.1037315 0.2039156 0.3942165
##  [6,] 0.05223945 0.1037052 0.2038821 0.3941690
##  [7,] 0.05216542 0.1036113 0.2037538 0.3939819
##  [8,] 0.05222557 0.1034160 0.2033340 0.3932838
##  [9,] 0.05782853 0.1054948 0.2031468 0.3912976
## [10,] 0.12268242 0.1489429 0.2246741 0.3961455
---
title: "Solving a parabolic equation"
date: "Created: 20-04-2025. Last modified: `r format(Sys.time(), '%d-%m-%Y.')`"
output:
  html_document:
    mathjax: "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
    highlight: pygments
    theme: flatly
    code_folding: hide # class.source = "fold-hide" to hide code and add a button to show it
    df_print: paged
    # toc: true
    # toc_float:
    #   collapsed: true
    #   smooth_scroll: true
    number_sections: false
    fig_caption: true
    code_download: true
always_allow_html: true
bibliography: 
  - references.bib
  - grateful-refs.bib
header-includes:
  - \newcommand{\ar}{\mathbb{R}}
  - \newcommand{\llav}[1]{\left\{#1\right\}}
  - \newcommand{\pare}[1]{\left(#1\right)}
  - \newcommand{\Ncal}{\mathcal{N}}
  - \newcommand{\Vcal}{\mathcal{V}}
  - \newcommand{\Ecal}{\mathcal{E}}
  - \newcommand{\Wcal}{\mathcal{W}}
---

```{r xaringanExtra-clipboard, echo = FALSE}
htmltools::tagList(
  xaringanExtra::use_clipboard(
    button_text = "<i class=\"fa-solid fa-clipboard\" style=\"color: #00008B\"></i>",
    success_text = "<i class=\"fa fa-check\" style=\"color: #90BE6D\"></i>",
    error_text = "<i class=\"fa fa-times-circle\" style=\"color: #F94144\"></i>"
  ),
  rmarkdown::html_dependency_font_awesome()
)
```


```{css, echo = FALSE}
body .main-container {
  max-width: 100% !important;
  width: 100% !important;
}
body {
  max-width: 100% !important;
}

body, td {
   font-size: 16px;
}
code.r{
  font-size: 14px;
}
pre {
  font-size: 14px
}
.custom-box {
  background-color: #f5f7fa; /* Light grey-blue background */
  border-color: #e1e8ed; /* Light border color */
  color: #2c3e50; /* Dark text color */
  padding: 15px; /* Padding inside the box */
  border-radius: 5px; /* Rounded corners */
  margin-bottom: 20px; /* Spacing below the box */
}
.caption {
  margin: auto;
  text-align: center;
  margin-bottom: 20px; /* Spacing below the box */
}
```


Let us set some global options for all code chunks in this document.


```{r}
# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = TRUE,    
  # Disable warnings printed by R code chunks
  warning = TRUE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
# Start the figure counter
fig_count <- 0
# Define the captioner function
captioner <- function(caption) {
  fig_count <<- fig_count + 1
  paste0("Figure ", fig_count, ": ", caption)
}
# Define the function to truncate a number to two decimal places
truncate_to_two <- function(x) {
  floor(x * 100) / 100
}

m1table <- rSPDE:::m1table
m2table <- rSPDE:::m2table
m3table <- rSPDE:::m3table
m4table <- rSPDE:::m4table
```


```{r}
# install.packages("INLA",repos=c(getOption("repos"),INLA="https://inla.r-inla-download.org/R/testing"), dep=TRUE)
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)
library(inlabru)
library(rSPDE)
library(MetricGraph)
library(grateful)

library(ggplot2)
library(reshape2)
library(plotly)
```


# Utilitary functions

```{r}
# For each m and beta, this function returns c_m/b_{m+1} and the roots of rb and rc
my.get.roots <- function(order, beta) {
  mt <- get(paste0("m", order, "table"))
  rb <- rep(0, order + 1)
  rc <- rep(0, order)
  if(order == 1) {
      rc = approx(mt$beta, mt[[paste0("rc")]], beta)$y
    } else {
      rc = sapply(1:order, function(i) {
        approx(mt$beta, mt[[paste0("rc.", i)]], beta)$y
      })
    }
    rb = sapply(1:(order+1), function(i) {
      approx(mt$beta, mt[[paste0("rb.", i)]], xout = beta)$y
    })
    factor = approx(mt$beta, mt$factor, xout = beta)$y
  return(list(rb = rb, rc = rc, factor = factor))
}

# Function the polynomial coefficients in increasing order like a+bx+cx^2+...
poly.from.roots <- function(roots) {
  
  coef <- 1
  for (r in roots) {coef <- convolve(coef, c(1, -r), type = "open")}
  return(coef)
}
# Function to compute the partial fraction parameters
compute.partial.fraction.param <- function(factor, pr_roots, pl_roots, cte) {
  
  pr_coef <- c(0, poly.from.roots(pr_roots)) 
  pl_coef <- poly.from.roots(pl_roots) 
  factor_pr_coef <- pr_coef
  pr_plus_pl_coef <- factor_pr_coef + cte/factor * pl_coef
  res <- gsignal::residue(factor_pr_coef, pr_plus_pl_coef)
  return(list(r = res$r, p = res$p, k = res$k)) 
}
# Function to compute the fractional operator
my.fractional.operators.frac <- function(L, beta, C, scale.factor, m = 1, time_step) {
  
  C <- Matrix::Diagonal(dim(C)[1], rowSums(C)) 
  Ci <- Matrix::Diagonal(dim(C)[1], 1 / rowSums(C)) 
  I <- Matrix::Diagonal(dim(C)[1])
  L <- L / scale.factor 
  LCi <- L %*% Ci
  if(beta == 1){
    L <- L * scale.factor^beta
    return(list(Ci = Ci, C = C, LCi = LCi, L = L, m = m, beta = beta, LHS = C + time_step * L))
  } else {
    roots <- my.get.roots(m, beta)
    scaling <- scale.factor^beta
    poles_rs_k <- compute.partial.fraction.param(roots$factor, roots$rc, roots$rb, time_step)

    partial_fraction_terms <- list()
    for (i in 1:(m+1)) {partial_fraction_terms[[i]] <- (LCi - poles_rs_k$p[i] * I)/poles_rs_k$r[i]}
    partial_fraction_terms[[m+2]] <- ifelse(is.null(poles_rs_k$k), 0, poles_rs_k$k) * I
    return(list(Ci = Ci, C = C, LCi = LCi, L = L, m = m, beta = beta, partial_fraction_terms = partial_fraction_terms))
  }
}
# Function to solve the iteration
my.solver.frac <- function(obj, v){
  beta <- obj$beta
  m <- obj$m
  C <- obj$C
  Ci <- obj$Ci
  if (beta == 1){
    return(solve(obj$LHS, v))
  } else {
    partial_fraction_terms <- obj$partial_fraction_terms
    output <- partial_fraction_terms[[m+2]] %*% v
    for (i in 1:(m+1)) {output <- output + solve(partial_fraction_terms[[i]], v)}
    return(Ci %*% output)
  }
}
# Function to build a tadpole graph and create a mesh
gets.graph.tadpole <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 10000)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges <- list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$set_manual_edge_lengths(edge_lengths = c(1,2))
  graph$build_mesh(h = h)
  return(graph)
}
# Function to compute the eigenfunctions 
tadpole.eig <- function(k,graph){
x1 <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[graph$mesh$PtE[,1]==1,2]) 
x2 <- c(0,graph$get_edge_lengths()[2]*graph$mesh$PtE[graph$mesh$PtE[,1]==2,2]) 

if(k==0){ 
  f.e1 <- rep(1,length(x1)) 
  f.e2 <- rep(1,length(x2)) 
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  f = list(phi=f1/sqrt(3)) 
  
} else {
  f.e1 <- -2*sin(pi*k*1/2)*cos(pi*k*x1/2) 
  f.e2 <- sin(pi*k*x2/2)                  
  
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  
  if((k %% 2)==1){ 
    f = list(phi=f1/sqrt(3)) 
  } else { 
    f.e1 <- (-1)^{k/2}*cos(pi*k*x1/2)
    f.e2 <- cos(pi*k*x2/2)
    f2 = c(f.e1[1],f.e2[1],f.e1[-1],f.e2[-1]) 
    f <- list(phi=f1,psi=f2/sqrt(3/2))
  }
}
return(f)
}
# Function to order the vertices for plotting
plotting.order <- function(v, graph){
  edge_number <- graph$mesh$VtE[, 1]
  pos <- sum(edge_number == 1)+1
  return(c(v[1], v[3:pos], v[2], v[(pos+1):length(v)], v[2]))
}
# Function to plot in 3D
graph.plotter.3d <- function(graph, 
                             U_true, 
                             U_approx1, 
                             #U_approx2, 
                             time_seq, 
                             time_step){
  x <- graph$mesh$V[, 1]
  y <- graph$mesh$V[, 2]
  x <- plotting.order(x, graph)
  y <- plotting.order(y, graph)
  weights <- graph$mesh$weights
  
  cumsum1 <- sqrt(time_step * cumsum(t(weights) %*% (U_true - U_approx1)^2))
  #cumsum2 <- sqrt(time_step * cumsum(t(weights) %*%(U_true - U_approx2)^2))
  #cumsum3 <- sqrt(time_step * cumsum(t(weights) %*% (U_approx1 - U_approx2)^2))
  
  U_true <- apply(U_true, 2, plotting.order, graph = graph)
  U_approx1 <- apply(U_approx1, 2, plotting.order, graph = graph)
  #U_approx2 <- apply(U_approx2, 2, plotting.order, graph = graph)
  
  plot_data <- data.frame(
  x = rep(x, times = ncol(U_true)),
  y = rep(y, times = ncol(U_true)),
  z_true = as.vector(U_true),
  z_approx1 = as.vector(U_approx1),
  #z_approx2 = as.vector(U_approx2),
  frame = rep(time_seq, each = length(x)))
  
  x_range <- range(x)
  y_range <- range(y)
  z_range <- range(c(U_true, 
                     #U_approx2, 
                     U_approx1))
  
  p <- plot_ly(plot_data, frame = ~frame) %>%
  add_trace(x = ~x, y = ~y, z = ~z_true, type = "scatter3d", mode = "lines", name = "T",
            line = list(color = "green", width = 2)) %>%
  add_trace(x = ~x, y = ~y, z = ~z_approx1, type = "scatter3d", mode = "lines", name = "A1",
            line = list(color = "red", width = 2)) %>%
  # add_trace(x = ~x, y = ~y, z = ~z_approx2, type = "scatter3d", mode = "lines", name = "A2", 
  #           line = list(color = "blue", width = 2)) %>%
  layout(
    scene = list(
      xaxis = list(title = "x", range = x_range),
      yaxis = list(title = "y", range = y_range),
      zaxis = list(title = "z", range = z_range),
      aspectratio = list(x = 2.4, y = 1.2, z = 1.2),
      camera = list(eye = list(x = 1.5, y = 1.5, z = 1), center = list(x = 0, y = 0, z = 0))),
    updatemenus = list(list(type = "buttons", showactive = FALSE,
                            buttons = list(
          list(label = "Play", method = "animate",
               args = list(NULL, list(frame = list(duration = 100, redraw = TRUE), fromcurrent = TRUE))),
          list(label = "Pause", method = "animate",
               args = list(NULL, list(mode = "immediate", frame = list(duration = 0), redraw = FALSE)))
        )
      )
    ),
    title = "Time: 0"
  ) %>% plotly_build()
  for (i in seq_along(p$x$frames)) {
    t <- time_seq[i]
    err1 <- signif(cumsum1[i], 6)
    #err2 <- signif(cumsum2[i], 4)
    #err3 <- signif(cumsum3[i], 4)
    p$x$frames[[i]]$layout <- list(title = paste0("Time: ", t, 
                                                  # " | cum E=T-A2: ", err2, 
                                                  # " | cum E=A1-A2: ", err3,
                                                  " | cum E=T-A1: ", err1))
  }
  return(p)
}
# Function to plot the error at each time step
error.at.each.time.plotter <- function(graph, U_true, U_approx1, U_approx2, time_seq, time_step) {
  weights <- graph$mesh$weights
  error_at_each_time1 <- t(weights) %*% (U_true - U_approx1)^2
  error_at_each_time2 <- t(weights) %*% (U_true - U_approx2)^2
  error_between_both_approx <- t(weights) %*% (U_approx1 - U_approx2)^2
  
  error1 <- sqrt(as.double(t(weights) %*% (U_true - U_approx1)^2 %*% rep(time_step, ncol(U_true))))
  error2 <- sqrt(as.double(t(weights) %*% (U_true - U_approx2)^2 %*% rep(time_step, ncol(U_true))))
  errorb <- sqrt(as.double(t(weights) %*% (U_approx1 - U_approx2)^2 %*% rep(time_step, ncol(U_true))))
  
  fig <- plot_ly() %>% 
  add_trace(
  x = ~time_seq, y = ~error_at_each_time1, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'red', width = 2),
  marker = list(color = 'red', size = 4),
  name = paste0("E=T-A1: ", sprintf("%.3e", error1))
) %>% 
  add_trace(
  x = ~time_seq, y = ~error_at_each_time2, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'blue', width = 2, dash = "dot"),
  marker = list(color = 'blue', size = 4),
  name = paste0("E=T-A2: ", sprintf("%.3e", error2))
) %>% 
  add_trace(
  x = ~time_seq, y = ~error_between_both_approx, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'orange', width = 2),
  marker = list(color = 'orange', size = 4),
  name = paste0("E=A1-A2: ", sprintf("%.3e", errorb))
) %>% 
  layout(
  title = "Error at Each Time Step",
  xaxis = list(title = "Error at Each Time Step"),
  yaxis = list(title = "Error"),
  legend = list(x = 0.1, y = 0.9)
)
  return(fig)
}
# Function to compute the eigenvalues and eigenfunctions
gets.eigen.params <- function(N_finite = 4, kappa = 1, alpha = 0.5, graph){
  EIGENVAL_ALPHA <- NULL
  EIGENFUN <- NULL
  for (j in 0:N_finite) {
      lambda_j_alpha <- (kappa^2 + (j*pi/2)^2)^(alpha/2)
      e_j <- tadpole.eig(j,graph)$phi
      EIGENVAL_ALPHA <- c(EIGENVAL_ALPHA, lambda_j_alpha)         
      EIGENFUN <- cbind(EIGENFUN, e_j)
      if (j>0 && (j %% 2 == 0)) {
        lambda_j_alpha <- (kappa^2 + (j*pi/2)^2)^(alpha/2)
        e_j <- tadpole.eig(j,graph)$psi
        EIGENVAL_ALPHA <- c(EIGENVAL_ALPHA, lambda_j_alpha)         
        EIGENFUN <- cbind(EIGENFUN, e_j)
      }
  }
  return(list(EIGENVAL_ALPHA = EIGENVAL_ALPHA,
              EIGENFUN = EIGENFUN))
}
```


We want to solve the fractional diffusion equation
\begin{equation}
\label{eq:maineq}
    \partial_t u+(\kappa^2-\Delta_\Gamma)^{\alpha/2} u=f \text { on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma,
\end{equation}
where $u$ satisfies the Kirchhoff vertex conditions
\begin{equation}
\label{eq:Kcond}
    \left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V: \sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\}
\end{equation}
The solution is given by
\begin{equation}
\label{eq:sol_reprentation}
        u(s,t) = \displaystyle\sum_{j\in\mathbb{N}}e^{-\lambda^{\alpha/2}_jt}\left(u_0, e_j\right)_{L_2(\Gamma)}e_j(s) + \int_0^t \displaystyle\sum_{j\in\mathbb{N}}e^{-\lambda^{\alpha/2}_j(t-r)}\left(f(\cdot, r), e_j\right)_{L_2(\Gamma)}e_j(s)dr.
\end{equation}

If we choose $w_j$ and $v_j$ and take the initial condition and the right hand side funciton as 

\begin{equation}
    u_0(s) = \sum_{j=0}^{N} w_j e_j(s) \text{ and so } \left(u_0, e_j\right)_{L_2(\Gamma)} = w_j,
\end{equation}
\begin{equation}
   \text{In matrix notation: } \quad\boldsymbol{U}_0 = \boldsymbol{E}_h\boldsymbol{c}, \quad  \boldsymbol{E}_h = \left[e_0, e_1, \ldots, e_{N}\right], \quad \boldsymbol{w} = \left[w_0, w_1, \ldots, w_{N}\right]^\top,
\end{equation}
\begin{equation}
   \text{In } \texttt{R}: \texttt{U_0 <- EIGENFUN %*% coeff_U_0}
\end{equation}
and
\begin{equation}
    f(s,t) = \sum_{j=0}^{N} v_j e^{-\lambda^{\alpha/2}_jt} e_j(s) \text{ and so } \left(f(\cdot,r), e_j\right)_{L_2(\Gamma)} = v_j e^{-\lambda^{\alpha/2}_jr},
\end{equation}
\begin{equation}
   \text{In matrix notation: } \quad\boldsymbol{f} = \boldsymbol{E}_h \boldsymbol{V}, \quad \boldsymbol{V}_{ji} = v_je^{-\lambda^{\alpha/2}_jt_i}
\end{equation}
\begin{equation}
   \text{In } \texttt{R}: \texttt{FF <- EIGENFUN %*% (coeff_FF * exp(-outer(EIGENVAL_ALPHA, time_seq)))}
\end{equation}
then the solution is given by
\begin{align}
        u(s,t) &= \displaystyle\sum_{j=0}^{N}w_je^{-\lambda^{\alpha/2}_jt}e_j(s) + \int_0^t \displaystyle\sum_{j=0}^Ne^{-\lambda^{\alpha/2}_j(t-r)}v_j e^{-\lambda^{\alpha/2}_jr}e_j(s)dr\\
        &= \displaystyle\sum_{j=0}^{N}w_je^{-\lambda^{\alpha/2}_jt}e_j(s) + t \displaystyle\sum_{j=0}^Nv_j e^{-\lambda^{\alpha/2}_jt}e_j(s)\\
        &= \displaystyle\sum_{j=0}^{N}w_je^{-\lambda^{\alpha/2}_jt}e_j(s) + tf(s,t)\\
        &= \displaystyle\sum_{j=0}^{N}(w_j+tv_j)e^{-\lambda^{\alpha/2}_jt}e_j(s) 
\end{align}

\begin{equation}
   \text{In matrix notation: } \quad\boldsymbol{U} =\boldsymbol{E}_h \boldsymbol{W} + (\boldsymbol{t}\boldsymbol{f}^\top)^\top, \quad \boldsymbol{W}_{ji} = w_je^{-\lambda^{\alpha/2}_jt_i},\quad \boldsymbol{t} = \left[t_0, t_1, \ldots, t_{K}\right]^\top
\end{equation}
\begin{equation}
   \text{In } \texttt{R}: \texttt{U_true <- EIGENFUN %*% (coeff_U_0 * exp(-outer(EIGENVAL_ALPHA, time_seq)))+ t(time_seq * t(FF))}
\end{equation}

```{r}
kappa <- 1
alpha <- 0.5 # from 0.5 to 2
m = 4
beta <- alpha/2
N_finite = 4 # choose even
adjusted_N_finite <- N_finite + N_finite/2 + 1
# Coefficients for u_0 and f
coeff_U_0 <- 50*(1:adjusted_N_finite)^-1
coeff_U_0[-5] <- 0
coeff_FF <- rep(0, adjusted_N_finite)
coeff_FF[7] <- 10
T_final <- 2

overkill_time_power <- 9
overkill_h_power <- 9

time_steps <- 0.1 * 2^-c(overkill_time_power:0)
hs <- 0.1 * 2^-c(overkill_h_power:0)

overkill_time_step <- time_steps[1]
overkill_h <- hs[1]

overkill_time_seq <- seq(0, T_final, by = overkill_time_step)

overkill_graph <- gets.graph.tadpole(h = overkill_h)
overkill_graph$compute_fem()
overkill_weights <- overkill_graph$mesh$weights
overkill_eigen_params <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = overkill_graph)
EIGENVAL_ALPHA <- overkill_eigen_params$EIGENVAL_ALPHA
overkill_EIGENFUN <- overkill_eigen_params$EIGENFUN


overkill_U_true <- overkill_EIGENFUN %*% 
  outer(1:length(coeff_U_0), 1:length(overkill_time_seq), 
        function(i, j) (coeff_U_0[i] + coeff_FF[i] * overkill_time_seq[j]) * exp(-EIGENVAL_ALPHA[i] * overkill_time_seq[j]))

by_vector <- 2^c(0:overkill_time_power)

errors <- matrix(NA, nrow = length(hs), ncol = length(time_steps))
for (i in 1:length(hs)) {
  h <- hs[i]
  graph <- gets.graph.tadpole(h = h)
  graph$compute_fem()
  G <- graph$mesh$G
  C <- graph$mesh$C
  L <- kappa^2*C + G
  eigen_params <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = graph)
  EIGENFUN <- eigen_params$EIGENFUN
  U_0 <- EIGENFUN %*% coeff_U_0
  A <- graph$fem_basis(overkill_graph$get_mesh_locations())
  for (j in 1:length(time_steps)) {
    time_step <- time_steps[j]
    coarse_indices <- seq(1, length(overkill_time_seq), by = by_vector[j])
    time_seq <- overkill_time_seq[coarse_indices]
    my_op_frac <- my.fractional.operators.frac(L, beta, C, scale.factor = kappa^2, m = m, time_step)
    INT_BASIS_EIGEN <- t(overkill_EIGENFUN) %*% overkill_graph$mesh$C %*% A
    FF_approx <- t(INT_BASIS_EIGEN) %*% 
      outer(1:length(coeff_FF), 1:length(time_seq), 
        function(i, j) coeff_FF[i] * exp(-EIGENVAL_ALPHA[i] * time_seq[j]))
    U_approx <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
    U_approx[, 1] <- U_0
    for (k in 1:(length(time_seq) - 1)) {
      U_approx[, k + 1] <- as.matrix(my.solver.frac(my_op_frac, my_op_frac$C %*% U_approx[, k] + time_step * FF_approx[, k + 1]))
      }
    sliced_overkill_U_true <- overkill_U_true[, coarse_indices]
    projected_U_approx <- A %*% U_approx
    errors[i,j] <- sqrt(as.double(t(overkill_weights) %*% (sliced_overkill_U_true - projected_U_approx)^2 %*% rep(time_step, length(time_seq))))
  }
}
```

```{r}
graph.plotter.3d(overkill_graph, sliced_overkill_U_true, projected_U_approx, time_seq, time_step)
```

```{r, fig.align='center', fig.dim= c(10,10), fig.cap = captioner("Caption")}
df <- melt(errors)
colnames(df) <- c("h", "time_step", "Value")

df$h <- factor(df$h, levels = unique(df$h), labels = hs)
df$time_step <- factor(df$time_step, levels = unique(df$time_step), labels = time_steps)

p <- ggplot(df, aes(x = time_step, y = h, fill = Value, text = paste("Value:", Value))) +
  geom_tile(color = "black") +  # Add borders
  geom_text(aes(label = round(Value, 8)), size = 3) +  # Add value labels
  scale_fill_gradient(low = "white", high = "red") +
  coord_fixed() +
  theme_minimal()

ggplotly(p, tooltip = "text")


```
```{r}
print(errors)
```

